### Introduction

In this Section we shall apply the basic theory of z-transforms to help us to obtain the response or output sequence for a discrete system. This will involve the concept of the transfer function and we shall also show how to obtain the transfer functions of series and feedback systems. We will also discuss an alternative technique for output calculations using convolution. Finally we shall discuss the initial and final value theorems of z-transforms which are important in digital control.

#### Prerequisites

- be familiar with basic z-transforms, particularly the shift properties

#### Learning Outcomes

- obtain transfer functions for discrete systems including series and feedback combinations
- state the link between the convolution summation of two sequences and the product of their z-transforms

#### Contents

1 Applications of z-transforms1.1 Transfer (or system) function

1.2 Second order systems

1.3 Combinations of systems

2 Convolution and z-transforms

3 Initial and final value theorems of z-transforms

3.1 Initial value theorem

3.2 Final value theorem